Simplicial lists in operad theory I

TL;DR

This paper introduces a new categorical framework using simplicial lists to model non-symmetric operads and their higher-dimensional generalizations, providing new tools for operad theory and $ ext{infinity}$-operads.

Contribution

It defines the category of simplicial lists, constructs a nerve functor for operads, and develops a model for non-symmetric $ ext{infinity}$-operads, expanding operad theory tools.

Findings

Constructed a fully-faithful nerve functor from operads to simplicial lists.

Showed that the category of simplicial lists is a presheaf category.

Developed a coherent nerve functor for $ ext{infinity}$-operads.

Abstract

We define a category $\mathsf{List}$ whose objects are sets and morphisms are mappings which assign to an element in the domain an ordered sequence (list) of elements in the codomain. We introduce and study a category of simplicial objects $\mathsf{slist}$ whose objects are functors $\Delta^{op} \to \mathsf{List}$, which we call simplicial lists, and morphisms are natural transformations which have functions as components. We demonstrate that $\mathsf{sList}$ supports the combinatorics of (non-symmetric) operads by constructing a fully-faithful nerve functor $N^l : \mathsf{Operad} \to \mathsf{sList}$ from the category of operads. This leads to a reasonable model for the theory of non-symmetric $\infty$-operads. We also demonstrate that $\mathsf{sList}$ has the structure of a presheaf category. In particular, we study a subcategory $\mathsf{sList}_{\text{op}}$ of operadic simplicial lists, in which the nerve functor takes values. The latter category is also a presheaf category over a base whose objects may be interpreted as levelled trees. We construct a coherent nerve functor which outputs an $\infty$-operad for each operad enriched in Kan complexes. We also define homology groups of simplicial lists and study first properties.